Shinyashiki put the Rapids in front in the 29th minute. He took a deflected pass from teammate Michael Barrios deep in the Austin zone and beat Julio Cascante to the ball before pushing it between the legs of Austin keeper Brad Stuver and into the goal for the match's lone tally.
Rapids keeper William Yarbrough recorded two saves to record his seventh shutout of the season and the 11th of his two-year MLS career.
Austin FC has now gone 279 minutes without a goal dating back to the 81st minute of a 4-1 win over Portland on July 1.
Stuver was forced into two incredible saves over the final five minutes of the first half, both on one-on-one chances by Barrios, to keep the home side within reach. He finished with five saves in the loss.
The win allowed Colorado, which remains in fourth place in the West, to split the season-series. Austin FC beat the Rapids in April on the road in the teams' first meeting this season, recording the inaugural win in their MLS history.
Austin, in last place in the Western Conference, has won just twice in its 12 matches since that win at Colorado. Saturday's loss was its third straight while also being held scoreless for the third consecutive home match.
Austin has managed just six goals in its past 12 games, with four of the tallies coming in the victory over Portland. Austin, which is tied for the fewest goals in the league with 10, has failed to score in 10 of its first 15 games.
The Rapids won a road match against an expansion side for the first time since a 2011 victory at Vancouver.